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Transcript
Quadratic Equations
P.7
Definition of a Quadratic Equation
• A quadratic equation in x is an equation
that can be written in the standard form
• ax2 + bx + c = 0
• where a, b, and c are real numbers with a
not equal to 0. A quadratic equation in x is
also called a second-degree polynomial
equation in x.
The Zero-Product Principle
If the product of two algebraic expressions is
zero, then at least one of the factors is equal
to zero.
If AB = 0, then A = 0 or B = 0.
Solving a Quadratic Equation by
Factoring
1. If necessary, rewrite the equation in the form
ax2 + bx + c = 0, moving all terms to one side,
thereby obtaining zero on the other side.
2. Factor.
3. Apply the zero-product principle, setting each
factor equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
Text Example
• Solve 2x2 + 7x = 4 by factoring and then using the
zero-product principle.
Step 1 Move all terms to one side and obtain
zero on the other side. Subtract 4 from both sides
and write the equation in standard form.
2x2 + 7x - 4 = 4 - 4
2x2 + 7x - 4 = 0
Step 2 Factor.
2x2 + 7x - 4 = 0
(2x - 1)(x + 4) = 0
Solution cont.
• Solve 2x2 + 7x = 4 by factoring and then
using the zero-product principle.
Steps 3 and 4 Set each factor equal to
zero and solve each resulting equation.
2x-1=0
or x + 4 = 0
2x=1
x = -4
x = 1/2
Steps 5 check your solution
Example
(2x + -3)(2x + 1) = 5
4x2 - 4x - 3 = 5
4x2 - 4x - 8 = 0
4(x2-x-2)=0
4(x - 2)*(x + 1) = 0
x - 2 = 0, and x + 1 = 0
So x = 2, or -1
The Square Root Method
If u is an algebraic expression and d is a
positive real number, then u2 = d has exactly
two solutions.
If u2 = d, then u = d or u = -d
Equivalently,
If u2 = d then u = d
Completing the Square
If x2 + bx is a binomial then by adding (b/2) 2,
which is the square of half the coefficient of
x, a perfect square trinomial will result.
That is,
x2 + bx + (b/2)2 = (x + b/2)2
Text Example
What term should be added to the binomial x2
+ 8x so that it becomes a perfect square
trinomial? Then write and factor the
trinomial.
The term that should be added is the square of
half the coefficient of x. The coefficient of x
is 8. Thus, (8/2)2 = 42. A perfect square
trinomial is the result.
x2 + 8x + 42 = x2 + 8x + 16 = (x + 4)2
x2 + 6x = 7
B divided
by 2,
squared
x2 + 6x + 9 = 7 + 9
x + 3
2
= 16
x + 3 = 4
x = 4 - 3
x = 1, or - 7
Don’t forget
to add to both
sides
Quadratic Formula
- b  b - 4ac
x=
2a
2
x 2 - 8x + 5 = 0
x 2 - 8x + 5 = 0
- (-8)  ( -8) 2 - 4(1)(5)
x=
2(1)
8  64 - 20
x=
2
8  44
x=
2
8  2 11
x=
2
2( 4  11)
x=
2
x = 4  11
The Discriminant and the Kinds of Solutions
to ax2 + bx +c = 0
Discriminant
b2 – 4ac
Kinds of solutions
to ax2 + bx + c = 0
b2 – 4ac > 0
Two unequal real solutions
Graph of
y = ax2 + bx + c
Two x-intercepts
b2 – 4ac = 0
One real solution
(a repeated solution)
One x-intercept
b2 – 4ac < 0
No real solution;
two complex imaginary
solutions
No x-intercepts